Generalized heavy-to-light form factors in light-cone sum rules

Physics Letters B 730 (2014) 336–341

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Physics Letters B www.elsevier.com/locate/physletb

Generalized heavy-to-light form factors in light-cone sum rules Ulf-G. Meißner a,b , Wei Wang a,∗ a b

Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany

a r t i c l e

i n f o

Article history: Received 16 December 2013 Received in revised form 3 February 2014 Accepted 5 February 2014 Available online 13 February 2014 Editor: A. Ringwald

a b s t r a c t We study the form factors for a heavy meson into the S-wave K π /π π system with an invariant mass below 1 GeV. The mesonic final state interactions are described in terms of the scalar form factors, which are obtained from unitarized chiral perturbation theory. Employing generalized light-cone distribution amplitudes, we compute the heavy-to-light transition using light-cone sum rules. Our approach simultaneously respects constraints from analyticity and unitarity, and also takes advantage of the power expansion in the 1/mb and the strong coupling constant. © 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .

1. Introduction B decays into a light vector meson are of particular interest as they can provide valuable information to extract the Standard Model (SM) parameters and therefore test the SM. In the case that large deviations from the SM calculations are found, these will shed light on new physics scenarios. Examples for such type of decays include e.g. the process B → ρ (→ ππ )lν¯ for the extraction of the CKM matrix element | V ub |, the reaction B → K ∗ (→ K π )l+ l− to test the chirality structure in weak interaction, and the decay B s → J /ψφ(→ K K¯ ) to determine the B s – B¯ s mixing phase. Recent experimental data on these channels can be found in Refs. [1–4]. Due to the short lifetime, the light vector meson cannot be directly detected by experiments and must be reconstructed from the two or three pseudo-scalars π / K final state. Thus these decay modes are at least four-body processes and the semi-leptonic ones are refereed to as B l4 decays in the literature [5] (for a recent dispersion theoretical approach to this reaction, see Ref. [6]). To select candidate events and suppress the combinatorial background, experimentalists often implement kinematic cuts on the invariant mass. During this procedure various partial waves of the K π /ππ system may get entangled and bring dilutions to physical observables. Particularly it is very likely the S-wave contributions are of great importance [7–29]. Therefore it is mandatory to have reliable and accurate predictions considering the high precision achieved or to be achieved by experiments. Decay amplitudes for semi-leptonic B decays into two lightpseudo-scalar mesons show two distinctive features. On the one hand, the final state interaction of the two pseudo-scalars should

*

Corresponding author. E-mail address: [email protected] (W. Wang).

satisfy unitarity and analyticity. On the other hand, the b mass scale is much higher than the hadronic scale, which allows an expansion of the hard-scattering kernels in terms of the strong coupling constant and the dimensionless power-scaling parameter ΛQCD /mb . In this Letter, we aim to develop a formalism that makes use of both these advantages. It simultaneously combines the perturbation theory at the mb scale based on the operator product expansion and the low-energy effective theory inspired by the chiral symmetry to describe the S-wave ππ and K π scattering. For concreteness, we will choose the B → K π matrix elements with the K π invariant mass below 1 GeV as an example in the following, while other processes including the charm meson decay can be treated in an analogous way. If the factorization can be proved, these form factors will also play an important role in the study of charmless three-body B decays [30–33]. 2. Generalized form factor The matrix elements



   ( K π )0 ( p K π )s¯ γμ γ5 b B¯ ( p B )    m2 − m2 1 P μ − B 2 K π qμ F1B → K π m2K π , q2 = −i mK π

+ 

m2B − m2K π q2

q



qμ F0B → K π m2K π , q2





,

   ( K π )0 ( p K π )s¯ σμν qν γ5 b B¯ ( p B ) =−

F TB → K π (m2K π , q2 )  m K π (m B + m K π )





q2 P μ − m2B − m2K π qμ



(1)

define the S-wave generalized form factors Fi [16]. Here, P = p B + p K π and q = p B − p K π .

http://dx.doi.org/10.1016/j.physletb.2014.02.009 0370-2693/© 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .

U.-G. Meißner, W. Wang / Physics Letters B 730 (2014) 336–341

The K π system with invariant mass below 1 GeV can be treated as a light hadron and more explicitly in the kinematics region we are considering, the m K π is small and the K π system moves very fast, the soft-collinear effective theory (SCET) is applicable [34–37]. As shown later this K π system has similar light-cone distribution amplitudes with the ones for a light hadron. The transition matrix elements for B → K π may be factorized in the same way as the ordinary B-to-light ones like the B → π transition. It has been demonstrated in SCET that, in the soft contribution limit, the form factors obey factorization [37–39]:

 2

Fi = Ciξ q

+ Fi,

The hadronic representation of the correlation function consists in the contribution of the B meson and of the higher resonances and the continuum state:

Π HAD ( p K π , q) ( K π )0 ( p K π )| j Γ1 | B¯ ( p K π + q) B¯ ( p K π + q)| j Γ2 |0 = m2B − ( p K π + q)2

∞ ρ h (s, q 2 ) + ds , s − ( p K π + q)2

   ( K π )0 s¯ Γ b| B¯  ∝ F K π m2K π ,

where higher resonances and the continuum of states are described in terms of the spectral function ρ h (s, q2 ) and start from the threshold s0 . The correlation function in Eq. (5) can also be evaluated in the deep Euclidean region in QCD at the quark level. The quark–hadron duality guarantees the equality of the two calculations and thus we obtain the sum rules

       m2 ( K π )0 ( p K π ) j Γ1  B¯ ( p B ) B¯ ( p B ) j Γ2 |0 exp − B2 M

=

(3)

(4)

C X is an isospin factor and B 0 is proportional to the QCD condensate parameter. For the K − π + , C X = 1. Below the K + 3π threshold, about 911 MeV, the K π scattering is strictly elastic. The inelastic contributions in the K π scattering comes from the K + 3π or K η . In the region from 911 MeV to 1 GeV, the K + 3π channel has a limited phase space, and thus is generically suppressed. Moreover, as a process-dependent study, it has been demonstrated the states with two additional pions will not give sizeable contributions to physical observables [43]. Though differences may be expected, some similarities might be shared. We leave the K + 3π contributions for future work. The K η coupled-channel effects can be included in the unitarized approach of chiral perturbation theory [44–48]. In the following we will choose the light-cone sum rules (LCSR) to calculate the Fi . An analysis in other approaches like the k T factorization [49–53] would be similar, and for recent developments in this approach see Refs. [54–62]. As a reconciliation of the original QCD sum rule approach [63,64] and the application of perturbation theory to hard processes, LCSR exhibit several advantages in the calculation of quantities like the meson form factors [65–69]. In the hard scattering region the operator product expansion (OPE) near the light-cone is applicable. Based on the light-cone OPE, form factors are expressed as a convolution of light-cone distribution amplitudes (LCDA) with a perturbatively calculable hard kernel. The leading twist and a few sub-leading twist LCDA give the dominant contribution, while higher twist terms are suppressed. The calculation begins with the correlation function:

Π( p K π , q) = i

 





d4 x e iq·x ( K π )0 ( p K π ) T j Γ1 (x), j Γ2 (0) |0, (5)

where j Γ1 is one of the currents in Eq. (1) defining the form factors: j Γ1 = s¯ γμ γ5 b for F1 and F0 , and j Γ1 = s¯ σμν γ5 qν b for F T . ¯ γ5 d to interpolate the B meson, whose matrix We choose j Γ2 = bi element gives the decay constant f B :





B¯ ( p B )¯bi γ5 d|0 =

m2B mb + md

fB.

(6)

s0

1









ds exp −s/ M 2 Im Π QCD s, q2 .

π

(8)

(mb +ms )2

where the strangeness-changing scalar form factors are defined by

 0|¯sd| K π  = C X B 0 F K π m2K π .

(7)

s0

(2)

where C i are the short-distance and calculable functions, and ξ is a universal soft form factor from the large recoil symmetry in the heavy quark mb → ∞ and large energy E → ∞ limit [40]. Symmetry breaking terms, starting at order αs , can be encoded into  F i , and can be expressed as a convolution in terms of the LCDA [37–39,41,42]. Watson’s theorem implies that phases measured in the K π elastic scattering and in a decay channel where the K π system decouple with other hadrons are equal (modulo π radians). This leads to

337

In the above, a Borel transformation has been performed to improve the convergence of the OPE series, and to enhance the contribution of the low-lying states to the correlation function for suitably chosen values of M 2 . The calculation of Π QCD is based on the expansion of the Tproduct in the correlation function near the light-cone, which produces matrix elements of non-local quark–gluon operators. These quantities are in terms of the generalized LCDA of increasing twist [71–74]:

  1 ( K π )0 s¯ (x)γμ d(0)|0 = Np K πμ

mK π

  ( K π )0 s¯ (x)d(0)|0 = N   ( K π )0 s¯ (x)σμν d(0)|0

1

1

du e iup K π ·x Φ K π (u ),

0

du e iup K π ·x Φ Ks π (u ),

0

1

1

= − N ( p K π μ xν − p K π ν xμ ) 6

du e iup K π ·x Φ Kσ π (u ),

(9)

0

where N = C X B 0 F K π . Due to the Watson’s theorem, the above matrix elements are proportional to the K π scalar form factors which have been absorbed into the normalization constant N. As a result, s ,σ the distribution amplitudes, Φ K π and Φ K π , are real. The LCDA Φ K π is twist-2, and the other two are twist-3. Their normalizations are given as

1 du Φ K π (u ) =

ms − md mK π

0

1

1 du Φ Ks π (u )

0

,

du Φ Kσ π (u ) = 1.

=

(10)

0

The use of conformal symmetry in QCD [70] indicates that the twist-3 LCDA have the asymptotic form [71–74]:

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U.-G. Meißner, W. Wang / Physics Letters B 730 (2014) 336–341

Φ Ks π (u ) = 1, Φ Kσ π (u ) = 6u (1 − u ),

(11)

and the twist-2 LCDA can be expanded in terms of Gegenbauer moments:

Φ K π (u ) = 6u (1 − u )



3/2

an C n (2u − 1).

(12)

n

It is worthwhile to stress that these generalized LCDA for a twohadron system have the same form as the ones for a light meson [71–74]. 3. Results For the sake of presentation, we define



  Fi q2 , m2K π = C X B 0m K π F K π m2K π F¯ i m2K π , q2 ,

(13)

Fig. 1. Scalar K π form factors calculated in unitarized chiral perturbation theory. Solid, dashed and dotted lines correspond to the magnitude, the real and the imaginary part, in order.

with the expressions

F¯ + = N F

 1

du u

 exp −

¯ 2K π − uq ¯ mb2 + u um

2

u0 =

m2K π + q2 − s0 +



(m2K π + q2 − s0 )2 + 4m2K π (mb2 − q2 ) 2m2K π

uM 2

(17)

u0

 1 × −mb Φ K π (u ) + um K π Φ Ks π (u ) + m K π Φ Kσ π (u ) +

mb2

+q −u 2

2

m2K π

 m K π Φ Kσ π (u )

uM 2

6



+ exp −s0 / M F¯ − = N F

 1

du u

3

2

m K π Φ Kσ π (u 0 ) mb2 − u 20m2K π + q2 mb2 + u 20 m2K π − q2

6

 exp −

¯ 2K π − uq ¯ 2 mb2 + u um

 ,

(14)

 mb Φ K π ( u )

uM 2

u0

+ (2 − u )m K π Φ Ks π (u ) + − × −

u (mb2

+q −u 2

m K π Φ Kσ π (u )

2



3u

m K π Φ Kσ π (u )

m2K π ) + 2(mb2 u2 M 2

−q +u 2

2

Our formulae can be compared to the results for the B to a scalar q¯ q meson transition. Quantities including the invariant mass and LCDA for the K π system will be replaced by those for the scalar q¯ q resonance as in Refs. [10,75]. The scalar form factor F K π has been calculated in the unitarized approach embedded in the chiral perturbation theory, and we refer the reader to Ref. [16] for details. We quote these results displayed in Fig. 1, where the solid, dashed and dotted lines correspond to the magnitude, the real and the imaginary part of F K π , respectively. From this figure, we can see the imaginary part shows an approximate linear dependence on m2K π . Such behaviour can be derived from the calculation in chiral perturbation theory and we quote the next-to-leading order results [76]: χ

F K π (s) = 1 +

m2K π )

K K π ,K π = −

0

σ 2 m K π Φ K π (u 0 )

F¯ T = 2N F (m B + m K π )

+ mb



6

 1

2

du u

 exp −

(15)

, ¯ 2K π ) ¯ 2 + u um (mb2 − uq



6uM 2

m K π Φ Kσ π (u 0 )

exp[−s0 / M ]

6

mb2 − q2 + u 20 m2K π

2

where

mb + m s 2m2B f B

exp

m2B M2

 ,

μi = ¯J =

uM 2

 m K π Φ Kσ π (u )

+ mb



s 4 K π

(5μπ − 2μ K − 3μη8 )

8f2

1 32π

−  ,

(16)

1 8f2

M i2 32 2 f 2

π

2

log





2+

λ(s)  s



1

K K η8 , K π = −

u0

Φ K π (u )

+

(18)

where L r5 is a low energy constant, and

u 0 (mb2 + u 20 m2K π − q2 )

× −

f2

3

u 0 (mb2 + q2 − u 20 m2f ) + 2(mb2 − q2 + u 20 m2K π )



4L r5 s

1 + ¯J K π K K π , K π − ¯J K η8 K K η8 , K π ,

6

 × exp −s0 / M

NF =

1−u

.

2Σ − 5s +

 3s − 2Σ −



M i2

μ2

s

2K π s

 ,

 ,

,

M 12 − M 22 s





32K π

−

M 12 + M 22 M 12

−

log s + λ(s) + M 12 − M 22

M 22



 log

M 22 M 12

  + log s + λ(s) − M 12 + M 22 − log −s + λ(s) − M 12 + M 22   − log −s + λ(s) + M 12 − M 22 , (19)

2 and Σ = M π + M 2K ,  K π = M 2K − M π2 . f is the pion decay constant, f = 92.4 MeV, λ2 (s) = [s − ( M 1 + M 2 )2 ][s − ( M 1 + M 2 )2 ], and s ≡ s + i  ensures that the correct sheet of the logarithm is

U.-G. Meißner, W. Wang / Physics Letters B 730 (2014) 336–341

339

Fig. 2. The dependence of the form factor F¯ 1 (left panel) and the ratio of the continuum and total contributions (right panel) on the Borel parameter. Solid lines denote the central value while the dashed curves correspond to variations of threshold parameter: s0 = (34 ± 2) GeV2 . Results for F¯ 1 are stable when M 2 > 6 GeV2 , while the continuum contribution is typically smaller than 30%.

χ

set. The imaginary part of the scalar form factor F K π arises from the function ¯J :

Im[ ¯J ] =

1 λ(s) 16π

s

,

(20)

which leads to an approximate linear dependence on the m2K π below 1 GeV2 . However, this linear dependence disappears in the region with large m2K π since higher-order contributions become important and are taken into account in the unitarized approach. This has been discussed in detail in Ref. [16]. The B meson decay constant is taken from the Lattice QCD calculation of Ref. [77]: f B = (196.9 ± 8.9) MeV. As demonstrated above, one of the most key inputs is the two-hadron LCDA. We will use asymptotic forms for the twist-3 ones, but no knowledge on the twist-2 is available at present. In Ref. [78], the authors have studied the LCDA for the light scalar mesons below 1 GeV in the q¯ q scenario. We shall use these results in our numerical calculation, bearing in mind large uncertainties that may be introduced by this approximation. To the best of our knowledge, there is no available results on non-asymptotic twist-3 LCDA for scalar mesons below 1 GeV. Studies of LCDA for scalar mesons above 1 GeV can be found in Refs. [79–81], but these results are not applicable here due to the large differences in the (invariant) mass. In the future, we hope the situation can be improved using nonperturbative QCD tools including Lattice QCD simulations. It is interesting to notice that Fi can also be evaluated with other interpolating currents. One example is the chiral current [81, 82], which has the advantage of isolating different contributions by twists. In this framework, choosing the suitable current, one can completely smear out the uncertain twist-2 LCDA in the QCD calculation, with the price of the complex hadronic representation since the parity partner of the B meson also contributes to the same correlation function. The criteria in LCSR to find sets of parameters M 2 (the Borel parameter) and s0 (the continuum threshold) is that the resulting form factor does not depend much on the precise values of these parameters; additionally both the continuum contribution, the dispersive integral from s0 to ∞ in Eq. (7), and the higher power corrections, arising from the neglected higher twist LCDA, should not be significant. One more requirement on the s0 is that it should not be too much away from the “reasonable” value: s0 is to separate the ground state from higher mass contributions, and thus should be below the next known resonance, in this case, B 1 with J P = 1+ . Thus approximately this parameter should be close to 33 GeV2 [83]. Studies of ordinary heavy-to-light form factors

Fig. 3. The dependence of F¯ 1 on the squared momentum transfer q2 and the twohadron invariant mass square m2K π .

in LCSR, see for instance Ref. [84], also suggested a similar result, ranging from 33 GeV2 to 36 GeV2 , while some bigger values are derived in the recent update of B → π form factor in LCSR [85]. Numerical results based on LCSR for the auxiliary function F¯ 1 at the K π threshold m K π = m K + mπ are given in Fig. 2, where the dependence of the form factor F¯ 1 (left panel) and the continuum/total ratio (right panel) on the Borel parameter are shown. The continuum contribution to the form factors is obtained by invoking the quark–hadron duality above the threshold s0 and calculating the correlation function on the QCD side. Solid lines denote the central value while the dashed curves correspond to variations of threshold parameter: s0 = (34 ± 2) GeV2 . From this figure, we can see that results for F¯ 1 are stable against the variation of M 2 when M 2 > 6 GeV2 , and meanwhile the continuum contribution is typically smaller than 30%. Unfortunately, due to the lack of knowledge on the 3-particle twist-3 and higher twist generalized LCDA, we are unable to estimate the power corrections due to these LCDA, and we hope this situation can be improved with more dedicated studies in the future. Choosing the value M 2 = 8 GeV2 , we show the results in Fig. 3 for the dependence on the squared invariant mass of the K π system and the squared momentum transfer q2 . As we can see, the results increase with the q2 . This behaviour is similar to the B → π [85] and B → ρ [84] form factors. More results and phenomenological consequences will be published elsewhere.

340

U.-G. Meißner, W. Wang / Physics Letters B 730 (2014) 336–341

4. Conclusions We have formulated an approach to explore the S-wave generalized form factors for the heavy meson transitions into the ππ , K π final state. We have adopted unitarized chiral perturbation theory to account for the final state interactions, and include these effects in the scalar form factors and generalized light-cone distribution amplitudes. The heavy-to-light transition is calculated within QCD sum rules on the light-cone. Our approach simultaneously respects constraints from unitarity and analyticity, and also takes advantage of the power expansion in the 1/mb and the strong coupling constant. With these form factors at hand based on improved results on the generalized LCDA, one may reliably explore the S-wave effects in semi-leptonic heavy meson decays and further non-leptonic charmless three-body B processes if the factorization holds. Acknowledgements We thank Michael Döring, Feng-Kun Guo, Bastian Kubis and YuMing Wang for useful discussions. This work is supported in part by the DFG and the NSFC through funds provided to the SinoGerman CRC 110 “Symmetries and the Emergence of Structure in QCD”, the “EU I3HP Study of Strongly Interacting Matter” under the Seventh Framework Program of the EU. References [1] P. del Amo Sanchez, et al., BaBar Collaboration, Phys. Rev. D 83 (2011) 032007, arXiv:1005.3288 [hep-ex]. [2] A. Sibidanov, et al., Belle Collaboration, Phys. Rev. D 88 (2013) 032005, arXiv: 1306.2781 [hep-ex]. [3] R. Aaij, et al., LHCb Collaboration, J. High Energy Phys. 1308 (2013) 131, arXiv: 1304.6325 [hep-ex]. [4] R. Aaij, et al., LHCb Collaboration, Phys. Rev. D 87 (2013) 112010, arXiv: 1304.2600 [hep-ex]. [5] C.L.Y. Lee, M. Lu, M.B. Wise, Phys. Rev. D 46 (1992) 5040. [6] X.-W. Kang, B. Kubis, C. Hanhart, U.-G. Meißner, arXiv:1312.1193 [hep-ph]. [7] S. Stone, L. Zhang, Phys. Rev. D 79 (2009) 074024, arXiv:0812.2832 [hep-ph]. [8] S. Stone, L. Zhang, arXiv:0909.5442 [hep-ex]. [9] Y. Xie, P. Clarke, G. Cowan, F. Muheim, J. High Energy Phys. 0909 (2009) 074, arXiv:0908.3627 [hep-ph]. [10] P. Colangelo, F. De Fazio, W. Wang, Phys. Rev. D 81 (2010) 074001, arXiv: 1002.2880 [hep-ph]. [11] P. Colangelo, F. De Fazio, W. Wang, Phys. Rev. D 83 (2011) 094027, arXiv: 1009.4612 [hep-ph]. [12] O. Leitner, J.-P. Dedonder, B. Loiseau, B. El-Bennich, Phys. Rev. D 82 (2010) 076006, arXiv:1003.5980 [hep-ph]. [13] R. Fleischer, R. Knegjens, G. Ricciardi, Eur. Phys. J. C 71 (2011) 1832, arXiv: 1109.1112 [hep-ph]. [14] X. Liu, W. Wang, Y. Xie, arXiv:1309.0313 [hep-ph]. [15] C.-D. Lu, W. Wang, Phys. Rev. D 85 (2012) 034014, arXiv:1111.1513 [hep-ph]. [16] M. Döring, U.-G. Meißner, W. Wang, J. High Energy Phys. 1310 (2013) 011, arXiv:1307.0947 [hep-ph]. [17] U.-G. Meißner, W. Wang, J. High Energy Phys. 1401 (2014) 107, arXiv:1311.5420 [hep-ph]. [18] R.-H. Li, C.-D. Lu, W. Wang, Phys. Rev. D 83 (2011) 034034, arXiv:1012.2129 [hep-ph]. [19] D. Becirevic, A. Tayduganov, Nucl. Phys. B 868 (2013) 368, arXiv:1207.4004 [hep-ph]. [20] J. Matias, Phys. Rev. D 86 (2012) 094024, arXiv:1209.1525 [hep-ph]. [21] T. Blake, U. Egede, A. Shires, J. High Energy Phys. 1303 (2013) 027, arXiv: 1210.5279 [hep-ph]. [22] C. Bobeth, G. Hiller, D. van Dyk, Phys. Rev. D 87 (2013) 034016, arXiv: 1212.2321 [hep-ph]. [23] S. Jäger, J. Martin Camalich, J. High Energy Phys. 1305 (2013) 043, arXiv: 1212.2263 [hep-ph]. [24] S. Descotes-Genon, T. Hurth, J. Matias, J. Virto, J. High Energy Phys. 1305 (2013) 137, arXiv:1303.5794 [hep-ph]. [25] S. Faller, T. Feldmann, A. Khodjamirian, T. Mannel, D. van Dyk, arXiv:1310.6660 [hep-ph]. [26] Z.-H. Zhang, X.-H. Guo, Y.-D. Yang, arXiv:1308.5242 [hep-ph]. [27] I. Bediaga, J.M. de Miranda, Phys. Lett. B 633 (2006) 167, arXiv:hep-ex/0405019.

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